Problem 38 - Algebra and Trigonometry

Question

(a) use the Product-to-Sum Formulas to express each product as a sum, and (b) use the method of adding \(y\) -coordinates to graph each function on the interval \([0,2 \pi] .\) $$ h(x)=\cos (2 x) \cos (x) $$

Step-by-Step Solution

Answer

\( h(x) = \frac{1}{2} [\cos(3x) + \cos(x)] \). Graphing involves plotting this on \([0, 2\pi]\).

Step 1 - Identify the Product-to-Sum Formula

The formula for the product of two cosines is given by:\[\cos(A) \cos(B) = \frac{1}{2} [\cos(A + B) + \cos(A - B)]\]

Step 2 - Substitute the Angles into the Formula

Here, use the substitution where \(A = 2x\) and \(B = x\):\[h(x) = \cos(2x) \cos(x)\]

Step 3 - Apply the Product-to-Sum Formula

Applying the formula:\[\cos(2x) \cos(x) = \frac{1}{2} [\cos(2x + x) + \cos(2x - x)]\]

Step 4 - Simplify the Expression

Simplify inside the cosines:\[\cos(2x) \cos(x) = \frac{1}{2} [\cos(3x) + \cos(x)]\]

Step 5 - Conclusion (Express as a Sum)

Thus, the product \(\cos(2x) \cos(x)\) can be expressed as the sum:\[h(x) = \frac{1}{2} [\cos(3x) + \cos(x)]\]

Step 6 - Graphing the Function

To graph the function on the interval \([0, 2\pi]\), compute the values of \( h(x) \) using the sum formula. Calculate the y-coordinates for specific x-values and plot the points to draw the graph. Note that both \(\cos(3x)\) and \(\cos(x)\) have known wave patterns.

Key Concepts

trigonometric identities

Understanding trigonometric identities is essential when working with complex trigonometric expressions. These identities simplify the process by converting products into sums, making calculations easier and enhancing the understanding of function behaviors. The Product-to-Sum formulas are particularly useful. For instance, the formula for the product of two cosines is given by: \[ \text{cos}(A) \text{cos}(B) = \frac{1}{2} [\text{cos}(A + B) + \text{cos}(A - B)] \] This identity helps to transform an expression involving multiplication into a sum of cosines. This simplification is important for solving and graphing trigonometric equations.

graphing trigonometric functions

Graphing trigonometric functions requires understanding their basic shapes and transformations. For instance, the cosine function, \( \text{cos}(x) \), has a periodic wave pattern. To graph more complex functions like \( h(x) = \frac{1}{2} [\text{cos}(3x) + \text{cos}(x)] \), it's useful to break down the components:

• Graph \( \text{cos}(x) \). It oscillates between -1 and 1 with a period of \( 2\text{π} \).
• Graph \( \text{cos}(3x) \). This function has a higher frequency, oscillating three times faster than \( \text{cos}(x) \) within the same period.
• Combine the graphs by adding the y-coordinates of each function at corresponding x-values.

The combined graph may look more complex, but understanding the individual behaviors makes it easier to plot points and draw the overall curve. This method aids in visualizing how different trigonometric functions interact in a given interval.

cosine function transformations

Transformations of the cosine function involve changes in amplitude, period, and phase shift. Each of these transformations modifies the basic shape of \( \text{cos}(x) \):

• Amplitude: Multiplying \( \text{cos}(x) \) by a constant changes its peak height. For example, \( 2\text{cos}(x) \) doubles the height of the wave.
• Period: Changing the frequency involves multiplying the variable inside the cosine function. \( \text{cos}(3x) \) shortens the period, making the wave oscillate faster.
• Phase Shift: Adding or subtracting inside the argument of the cosine function shifts the wave horizontally. \( \text{cos}(x - \text{π}/2) \) moves the graph to the right by \( \text{π}/2 \).

Combining these transformations can model more complex behaviors. For example, in \( h(x) = \frac{1}{2} [\text{cos}(3x) + \text{cos}(x)] \), the addition of two cosine functions with different periods and amplitudes creates a wave with unique features. Understanding these transformations helps in analyzing and predicting the behavior of trigonometric functions.